Weak approximation results for quadratic forms in four variables

Let $F$ be a quadratic form in four variables, let $m\in \mathbb{N}$ and let $\mathbf{k}\in {\mathbb{Z}}^4$. We count integer solutions to $F(\mathbf{x})=0$ with $\mathbf{x}\equiv \mathbf{k}\mathrm{m}\mathrm{o}\mathrm{d}(m)$. One can compare this to the similar problem of counting solutions to $F(\mathbf{x})=0$ without the congruence condition. It turns out that adding the congruence condition sometimes gives a very different main term than the homogeneous case. In particular, there are examples where the number of primitive solutions to the problem is $0$, while the number of unrestricted solutions is nonzero.